Fibonacci sequences

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Virginia Polytechnic Institute

Early in the thirteenth century, Leonardo de Pisa, or, Fibonacci, introduced his famous rabbit problem, which may be stated simply as follows: assume that rabbits reproduce at a rate such that one pair is born each month from each pair of adults not less than two months old. If one pair is present initially, and if none die, how many pairs will be present after one year? The solution to the problem gives rise to a sequence {Un} known as the Classical Fibonacci Sequence. {Un} is defined by the recurrence relation

Un = Un-1 + Un-2, n ≥ 2, U₀ = 0, U₁ = 1

Many properties of this sequence have been derived.

A generalized sequence {Fn} can be obtained by retaining the law of recurrence and redefining the first two terms as F₁ = p', F₂ = p' + q' for arbitrary real numbers p' and q'. Moreover, by defining H₁ = p+iq, H₂ = r+is, p,q,r and s real, a complex sequence is determined. Hence, all the properties of the classical sequence can be extended to the complex case.

By reducing the classical sequence by a modulus m, many properties of the repeating sequence that results can be derived.

The Fibonacci sequence and associated golden ratio occur in communication theory, chemistry, and in nature.